SPECIAL SOLUTIONS TO A COMPACT EQUATION FOR DEEP-WATER GRAVITY WAVES
|
|
- Rodger Ford
- 5 years ago
- Views:
Transcription
1 SPECIAL SOLUTIONS TO A COMPACT EQUATION FOR DEEP-WATER GRAVITY WAVES FRANCESCO FEDELE AND DENYS DUTYKH ariv:.889v3 [physics.class-ph] Sep Abstract. Recently, Dyachenko & Zakharov () [9] have derived a compact form of the well known Zakharov integro-differential equation for the third order Hamiltonian dynamics of a potential flow of an incompressible, infinitely deep fluid with a free surface. Special traveling wave solutions of this compact equation are numerically constructed using the Petviashvili method. Their stability properties are also investigated. In particular, unstable traveling waves with wedge-type singularities, viz. peakons, are numerically discovered. To gain insights into the properties of these singular solutions, we also consider the academic case of a perturbed version of the compact equation, for which analytical peakons with exponential shape are derived. Finally, by means of an accurate Fouriertype spectral scheme it is found that smooth solitary waves appear to collide elastically, suggesting the integrability of the Zakharov equation. Contents. Introduction. Envelope dynamics and special solutions 3.. A Hamiltonian Dysthe equation.. Ground states and traveling waves 5 3. Traveling wave interactions Numerical method description Numerical results. Conclusions 3 Acknowledgements References 5. Introduction The Euler equations that describe the irrotational flow of an ideal incompressible fluid of infinite depth with a free surface are Hamiltonian. Their symplectic formulation was discovered by Zakharov (968) [38] in terms of the free-surface elevation η(x, t) and the velocity potential ϕ(x,t) = φ(x,z = η(x,t),t) evaluated at the free surface of the fluid. Here, η(x, t) and ϕ(x, t) are conjugated canonical variables with respect to the Hamiltonian Key words and phrases. water waves; deep water approximation; Hamiltonian structure; travelling waves; solitons. Corresponding author.
2 F. FEDELE AND D. DUTYKH H given by the total wave energy. It is well known that the Euler quations are completely integrable in several important limiting cases. For example, in a two-dimensional (-D) ideal fluid, unidirectional weakly nonlinear narrowband wave trains are governed by the Nonlinear Schrödinger (NLS) equation, which is integrable []. Integrability also holds for certain equations that models long waves in shallow waters, in particular the Korteweg de Vries (KdV) equation (see, for example, [,, 6, 36]) or the Camassa Holm (CH) equation []. For these equations, the associated Lax-pairs have been discovered and the Inverse Scattering Transform unveiled the dynamics of solitons, which elastically interact under the invariance of an infinite number of time-conserving quantities [,, 6, 36]. Another important limiting case of the Euler equations for the free-surface of an ideal flow is that considered by [38, 39]. By means of a third order expansion of H in the wave steepness, he derived an integro-differential equation in terms of canonical conjugate Fourier amplitudes, which has no restrictions on the spectral bandwidth. To derive the Zakharov (Z) equation, fast non-resonant interactions are eliminated via a canonical transformation that preserves the Hamiltonian structure [7, 39, 5]. The integrability of the Z equation is still an open question, but the fully nonlinear Euler equations are nonintegrable [7, 8]. Indeed, non-integrability can be easily proven by considering the terms of the perturbation series of the Hamiltonian in powers of the wave steepness limited on their resonant manifolds. To have non integrability, it is enough to prove that at least one of these amplitudes is nonzero. In this regard, [7, 8] conjectured that the Z equation for unidirectional water waves (-D) is integrable since the nonlinear fourth-order term of the Hamiltonian vanishes on the resonant manifold, leaving only trivial wave-wave interactions, which just cause nonlinear frequency shifts of the separate Fourier modes. They also pushed their analysis to the next order and proved that the effective fifth-order term does not vanish on the corresponding resonant manifold. Thus, the Euler equations of the free-surface hydrodynamics are not integrable in two dimensions [8]. Recently, Dyachenko& Zakharov()[9] realized that the trivial resonant quartetinteractions that occur in the -D free surface dynamics could be further removed by a canonical transformation. As a result, the Z equation drastically simplifies to the compact form ib t = Ωb+ i (b ( (b x ) ) 8 ( b x(b ) ) ) x x [ bk{ b x x } ( b x K{ b } ) ], (.) x where b is a canonical variable, b t, b x denote partial derivatives with respect to t and x and the symbols of the pseudo-differential operators Ω and K = H[ x ] (H being the Hilbert transform) are given, respectively, by g k and k, k being the Fourier transform parameter. Further, b relates to the wave surface η as ω η = a+c.c., (.) g where a is a conjugate canonical variable that relates to b via a non-trivial canonical transformation (see [9]), and c.c. denotes complex conjugation. To leading order in the wave steepness a = b.
3 SOLUTIONS TO A COMPACT DEEP-WATER EQUATION 3 In this study, we explore (.), hereafter referred to as cdz, for analytical studies and numerical investigations of special solutions in the form of solitary waves. The present paper is organized as follows. First, we derive the envelope equation associated to cdz and then smooth ground states and traveling waves are numerically computed using the Petviashvili method ([6], see also [8]). Further, unstable traveling waves with wedgetype singularities, viz. peakons, are also numerically identified. To gain insights into the properties of singular traveling waves we consider a perturbed version of the local compact equation, where the non-local terms involving the operator K are neglected in (.). For this academic case we are able to derive analytical peakons with exponential shape. Finally, the interaction of smooth traveling waves is numerically investigated by means of an accurate Fourier-type pseudo-spectral scheme.. Envelope dynamics and special solutions Consider the following ansatz for wave trains in deep water g b(,t) = ε a B(,T)e i(k x ω t), (.) ω where B is the envelope of the carrier wave e i(k x ω t), and = εk (x c g t), T = ε ω t, with k = ω g and ω as characteristic wavenumber and frequencies. The small parameter ε = k a is a characteristic wave steepness associated to an amplitude a and c g is the wave group velocity in deep water. From (.), the leading order wave surface η is given in terms of the envelope B as η(,t) = εa B(,T)e i(k x ω t) +c.c. (.) Using (.) in (.) yields the cdz envelope equation ib T = Ω ε B + i ( B S((SB) )+ib (SB) S ( B SB )) ε [ BK{ SB } S ( SBK{ B } )], (.3) where S = ε +i. The approximate dispersion operator Ω ε is defined as follows Ω ε := 8 + i 6 ε 5 8 ε + 7i 56 ε3, where o(ε 3 ) dispersion terms are neglected. Equation (.3) admits three invariants, viz. the action A, momentum M and the Hamiltonian H given, respectively, by [ H = B Ω ε B + i SB [B(SB) B SB] ε ] SB K( B ) d, (.) R A = R B Bd, M = R i ( B SB B(SB) ) d. (.5) Expanding the operator S in terms of ε, (.3) can be written in the form of a generalized derivative NLS equation as ib T = Ω ε B + B B 3iε B B ε BK{ B }+ε N (B)+ε 3 N 3 (B) =, (.6)
4 F. FEDELE AND D. DUTYKH where B denotes differentiation with respect to, and N (B) = 3 B (B ) +B B B B + B B + [ BK(B B BB )+B K B + ( BK B ) i N 3 (B) = i B B + i B (B B BB ) i BB B To leading order (.6) reduces to the NLS equation. ], [ BK B ( B K B ) ]... A Hamiltonian Dysthe equation. Hereafter, we will study a special case of (.6), whichisasymplecticversionofthetemporaldystheequation([], seealso[3,3,6,33]). Keeping terms to O(ε) in (.6) yields ib T = ( 8 + iε 6 ) B + B B 3iε B B εbk B, (.7) hereafter referred to as cdz-dysthe. Note that the original temporal Dysthe equation is not Hamiltonian since expressed in terms of multiscale variables, which are usually non canonical (see, for example, []). This is common in mathematical physics. Typically, the dynamics is governed by partial differential equations expressed in terms of physicallybased variables, which are not usually canonical. A transformation to new variables is needed in order to unveil the desired structure explicitly (see, for example, [7]). This is the case for the equations of motion for an ideal fluid: in the Eulerian description, they cannot be recast in a canonical form, whereas in a Lagrangian frame the Hamiltonian structure is revealed by Clebsch potentials (see, for example, [7, ]). Moreover, multiplescale perturbations of differential equations expressed in terms of non-canonical variables typically lead to approximate equations that do not maintain the fundamental conserved quantities, as the hydrostatic primitive equations on the sphere [], where energy and angular momentum conservation are lost under the hydrostatic approximation. As an example, consider the finite dimensional system of an harmonic oscillator in the classical canonical variables q(t) (coordinate) and p(t) (momentum). This admits the canonical form q = H p = p, ṗ = H q = q, (.8) where ṗ denotes time derivative, and H = (q + p )/. The flow in the phase-space is incompressible since the divergence vanishes: q q + ṗ p =. The transformation z = q +ip is canonical and (.8) transforms to ż = i H z = iz, ż = i H z = iz, (.9)
5 SOLUTIONS TO A COMPACT DEEP-WATER EQUATION 5 where H = z /, and z is the complex conjugate of z. It is straightforward to prove that the gauge transformation z = we iα w, with α as a free parameter, is also canonical, and (.9) remains unchanged in the new variables w and w. On the other hand, if one considers the coordinate change q Q =, P = p, (.) +αq then (.8) transforms to the noncanonical form P = Q ( +αp αp ), Q = P +αp. (.) This flow does not preserve volume as (.8) does, nonetheless equations (.) are those of a disguised harmonic oscillator obtained via the noncanonical change of variables (.) (see also []). The original Dysthe equation [] shares the same roots as (.). They both come from a noncanonical transformation of a Hamiltonian system. For the spatial version of the Dysthe equation, canonical variables have been identified by means of a gauge transformation, and the hidden Hamiltonian structure is unveiled []. The more general canonical transformation proposed by [9] yields the symplectic form (.7) of the Dysthe equation []. In the following, insights into the underlying dynamics of the cdz equation and associated Dysthe equation are to be gained if we construct some special families of solutions in the form of ground states and traveling waves of the envelope B, often just called solitons or solitary waves, as described below... Ground states and traveling waves. To begin, consider the cdz equation (.3). We construct numerically ground states and traveling waves (TWs) of the envelope B of the form B(,T) = F( ct)e iωt, where the function F( ) is in general complex, ω and c are dimensionless frequency and velocity of the TW with respect to a reference frame moving with the group velocity c g. In physical space the true frequency and velocity are given, respectively, by ω = ε ω ω and c = εcω /k. After substituting this ansatz into the governing equation (.3) we obtain the following nonlinear steady problem for F(in the moving frame ct) LF = N(F), where L = ω ic Ω ε and N(F) denotes nonlinear terms of the cdz of (.3). In order to solve this equation we use the Petviashvili method[6, 5, 8, 37]. This numerical approach has been successfully applied in deriving TWs of the spatial version of the Dysthe equation []. Schematically, the iteration takes the following form F n+ = S γ L N(F n ), S = F n,l F n F n,n(f n ), where S is the so-called stabilyzing factor and the exponent γ is usually defined as a function of the degree of nonlinearity p (p = 3 for the cdz equation). The rule of thumb prescribes the following formula γ = p p. The scalar product is defined in the L space.
6 6 F. FEDELE AND D. DUTYKH The inverse operator L can be efficiently computed in the Fourier space. To initialize the iterative process, one can use the analytical solution to the leading order NLS equation (see the associated Dysthe equation (.7)). We point out that this method can be very efficiently implemented using the Fast Fourier Transform (FFT) (see, for example, [3]). Without loosing generality, hereafter we just consider the leading term of the dispersion operator, viz. Ω ε = 8, since the soliton shape is only marginally sensitive to the higher order dispersion terms as shown in Figure. We also observed that the Petviashvili method does not converge if we retain the full dispersion operator. Figure shows the action A of the ground states (c =, but moving with the group velocity in the physical frame) as a function of the frequency ω computed via the Petviashvili iteration and numerical continuation for ε =.. The stability of a ground state is investigated numerically by means of an accurate Fourier-type spectral scheme [3, 3], see also []. We found that smooth ground states are stable in agreement with the criterion formulated by [3], since da > (see also [, 37]). Further, we notice that an abrupt reduction in the action A dω occursatacriticalfrequency ω c (ε)andsolitonswithwedge-type singularities, viz. peakons, bifurcate from a smooth solitary waves as shown in the right panel of Figure. Clearly, this bifurcation can be interpreted as a possible indication of the non-existence of smooth solitons above the critical threshold ω c. As one can see, as ω increases the soliton shape tends to become asymmetric andsteeper until thecritical threshold ω c =.85, above which the smooth solitary waves bifurcate to peakons, which are unstable in agreement with [3], since da <. Peakons also bifurcate from smooth solitons of given frequency ω as the dω steepness ε increases, as clearly seen in Figure 3. Note that in both cases ground states grow asymmetrically before bifurcating to smaller peakons, and so do travelling waves (c ). However, such bifurcation is not observed in the -D NLS or Dysthe equations (see, for example, []), as clearly seen in Figures 5 7 where we report the dependence of thethreeinvariants A, MandHontheparameters ω, candεforthecdzandcdz-dysthe equations respectively. Note that stable and elastic peakons have been discovered in a special limit of the integrable CH equation []. On the contrary, from numerical investigations of the cdz equation peakons appear unstable. The left panel of Figure shows the numerically converged peakon obtained via the Petviashvili scheme using N.5 6 Fourier modes for ε =.. As the frequency ω increases, peakons bifurcate at smaller steepness ε as clearly seen in the right panel of Figure. Indeed, for ground states we observed numerically that ω c (ε) ε (see left panel of Figure 8), and the same scaling holds also for travelling waves (c ). In the physical frame (x,t), owing to the scaling T = ε ω t, bifurcation occurs at a well defined frequency ω = ε ω c.37ω, which is independent of ε. Further, in the domain both the peakon amplitude a p = B = and the associated slope of the wedge singularity (corner) s p = a p B = = cot(θ/)
7 SOLUTIONS TO A COMPACT DEEP-WATER EQUATION 7 scales as ε 3/, θ being the interior angle of the peakon s singularity (see central and right panels of Figure 7). In the physical frame x, from (.) the slope s p becomes s p = εa B = x = ε 3 a p s p cot(7 /), which is independent of ε and corresponds to an angle θ 7. For a meaningful comparison of this result with the almost-highest solitary wave theory of Longuet-Higgins and Fox [,, 9], which predicts an angle of θ, higher order corrections to (.) hidden within the full canonical transformation of [9] should be accounted for. This is currently under study and will be discussed elsewhere. However, it is also important to mention that the cdz equation is derived assuming weak nonlinearities, so the strongly nonlinear peakon could be a non-physical artifact, whereas the almost-highest wave satisfies the full Euler equations and thus accounts for nonlinearities of all orders. The derivation of an analytical solution for peakons of the non-local cdz equation (.6) is a challenge. To gain more insights into the properties of these singular solutions we consider the academic case of a perturbed version of the local cdz, viz. ib T = 8 B + i ( B S((SB) )+(+δ)ib (SB) S ( B SB )), (.) Here, the non-local terms are dropped simply because peakons with exponential shape can be found analytically as function of the free parameter δ. Indeed, the ansatz satisfies (.) for B(,T) = a p e i(κ ωpt) e α c pt, (.3) κ = 8+δ 8ε, α = δ 8ε, ω p = +δ 3ε, c p = 8+δ 3ε. (.) The amplitude a p is still unknown and it rules the existence of peakons. Indeed, they arise as a balance between the dispersion B 8 and the O(ε ) term of the nonlinearity i B S((SB) ) in (.). These two terms are interpreted in distributional sense because they give rise to Dirac delta functions that must vanish by properly chosing the amplitude a p, thus satisfying the differential equation (.) in the sense of distributions. As a result a p = ε δ. (.5) Clearly, peakons exist for δ < and for δ = they reduce to periodic wave solutions of the local cdz. Thus, peakons can bifurcate from periodic waves of the local cdz as δ changes sign, and ground states occur for δ = 8. We note that both the NLS and Dysthe equations do not have higher nonlinear terms similar to i B S((SB) ) and thus they cannot support peakons. We conclude that wedge-type singular solutions are special features of the cdz equation. The left panel of Figure 9 shows the remarkable agreement of a numerical peakon (δ = 3/ and ε =.3) computed using the Petviashvili scheme (Fourier modes N.5 6 ) and the corresponding analytical solution (.3). Moreover, the amplitude of the numerical peakon agrees with the associated theoretical counterpart (.5) asfunction of ε (see right panel of Figure9). Finally, note that theanalytical peakon (.3) has a slope that scales as ε, in contrast to the ε 3/ scaling observed numerically
8 8 F. FEDELE AND D. DUTYKH Bj nd, j= 3 rd, j=3 th, j= 5 th, j=5 3 3 B j B /max( B ) 3 j=3 j= j=5 3 3 Figure. cdz equation: Convergence of the ground state B j (ε =.9, ω =., c = ) with respect to the approximate dispersion relation that retains terms up to second, third, fourth and fifth orders (j =,...,5). Left: soliton shapes; right: normalized differences Bj B for j = 3,,5. A A B C D ω E Re(B) D,ω =.75,ε =. C,ω =.5 B,ω =.3 A,ω =.5 E,ω = Figure. cdz equation: (left) Action A of ground states of the envelope B as function of the frequency ω and (right) bifurcation of a peakon from smooth ground states (ε =., c = ). for the full cdz singular solution. In the physical frame x, the interior angle of the wedge singularity is independent of ε and so is the peakon frequency ω p, in agreement with the numerical cdz peakons. This is an indirect evidence of the validity of our numerical results for the cdz equation. Finally, we note that as ε approaches zero, (.) tends to the NLS equation, and all the peakon parameters approach infinite values as an indication of the non-existence of singular solutions in the NLS limit. 3. Traveling wave interactions Hereafter, we investigate the collision of smooth traveling waves of the cdz equation (.3) by means of a highly accurate Fourier-type pseudo-spectral method. We will first briefly describe the adopted spectral approach and then discuss the numerical results.
9 SOLUTIONS TO A COMPACT DEEP-WATER EQUATION 9.8 A B C.8.6 A,ε =.,ω =. A.6.. D ε Re(B).. B,ε =.8 C,ε =.5 D,ε = Figure 3. cdz equation: (left) Action A of ground states of the envelope B as function of the steepness ε and (right) bifurcation of a peakon from smooth ground states (ω =., c = ) ω = B..5 A ε Figure. cdz peakon: (left) numerical solution via the Petviashvili scheme (ε =., c =, number of Fourier modes N.5 6 ) and (right) action of ground states of the envelope B as function of the steepness ε for different values of the frequency ω. 3.. Numerical method description. We rewrite (.3) in the operator form B t +L B = N(B), (3.) where L = i 8 and N(B) = ( B S((SB) )+ib (SB) S ( B SB )) + εi [ BK{ SB } S ( SBK{ B } )]. We solve (3.) numerically by applying the Fourier transform in the space variable. The transformed functions will be denoted by ˆB = F{B}. We recall that the symbol of the non-local term is equal to k, and that of L is i 8 k, k being the Fourier transform parameter. The nonlinear terms are computed in physical space, while space derivatives and the nonlocal operator K = H[ x ] are computed in Fourier space. For example, the
10 F. FEDELE AND D. DUTYKH.8.6 cdz A.8.6 cdz-dysthe M.5.5 H ω ω.5.5 ω Figure 5. cdz equation: Action A, momentum M and Hamiltonian H dependence onthefrequency ω forgroundstates oftheenvelope B (ε =.3, c = ) cdz 5.5 A.5 M 3 H cdz-dysthe ε...3. ε ε Figure 6. cdz equation: Action A, momentum M and Hamiltonian H dependence on the steepness parameter ε for ground states of the envelope B (ω =.8, c = ). term B B is discretized as F{B B } = F{ ( F (ˆB) ) F {ik ˆB}}, and all nonlinear terms are treated in a similar way. We note that the usual /3 rule is applied for anti-aliasing since we have to deal with cubic nonlinearities [3, 5, ]. In order to improve the stability of the space discretization procedure, we can integrate exactly the linear terms. This is achieved by making a change of variables [3, ]: { } Ŵ t = e (t t )L N e (t t )L Ŵ, Ŵ(t) e (t t )L ˆB(t), Ŵ(t ) = ˆB(t ).
11 SOLUTIONS TO A COMPACT DEEP-WATER EQUATION cdz A 3.5 M H.5 cdz-dysthe c c c Figure 7. cdz equation: Action A, momentum M and Hamiltonian H dependence on the velocity c for travelling waves of the envelope B(ε =., ω =.9). wc.. sp. ω=. ω=..37ε.9.6ε.6.ε.5 ap ε.7.6 ε.7.6 ε Figure 8. cdz peakon: dependence of the critical frequency ω c, peakon slope s p and amplitude a p on the steepness parameter ε. This result is obtained for ground states of the envelope B. Note that the same scalings hold true also for travelling waves. Finally, the resulting system of ODEs is discretized in space by the Verner s embedded adaptive 9(8) Runge Kutta scheme [35]. The step size is chosen adaptively using the socalled Hb digital filter [8, 9] to meet the prescribed error tolerance, set as of the order of machine precision. 3.. Numerical results. In all the performed simulations the accuracy has been checked by following the evolution of invariants (.), (.5). From a numerical point of view the cdz equation becomes gradually stiffer as the steepness parameter ε increases, or if higher order dispersion terms are included. Consequently, the conservation of invariants H, A and M might be degraded. Nevertheless, even in the stiffest situations a decent accuracy
12 F. FEDELE AND D. DUTYKH B analytical peakon numerical solution δ =.5, ǫ =.3 ap δ =.5, ǫ =.3 Analytical amplitude numerical solution c pt ǫ Figure 9. Left: numerical peakon (solid) of the perturbed local cdz equation (.) and associated analytical solution (.3) (dash) for ε =.3, δ =.5; Right: amplitude of the numerical peakon against its theoretical counterpart (.5) as function of ε. Figure. cdz envelope equation: collision of two smooth travelling waves (ε =.).
13 SOLUTIONS TO A COMPACT DEEP-WATER EQUATION B Figure. cdz envelope equation: Initial () shape and () after collision of the two interacting traveling waves of Figure. was assured by choosing a sufficiently large number of Fourier modes and the dispersion operator Ω ε = 8. For example, for ε =. the number N = 638 of Fourier modes were sufficient to achieve conservation of theinvariants close to 3. As anapplication, consider the interaction between two solitary waves traveling in opposing directions with the same speed (ε =.). The plot of Figure shows that the two solitons emerge out of the collision with the same shape, but a slight phase shift. The interaction appears elastic as clearly seen from the plot of Figure, which reports the initial and final shapes of the two solitary waves. Further, the interaction of two traveling waves with a ground state appears also elastic (see Figures 3). This suggests the integrability of the cdz equation (.3) in agreement with the recent results of []. However, the associated Hamiltonian version of the Dysthe equation (.7) does not support elastic collisions as shown in Figure.. Conclusions Special solutions of the compact Zakharov equation in the form of traveling waves are numerically constructed using the Petviashvili method. The stability of ground states agrees with the Vakhitov-Kolokolov criterion. Further, unstable ground states with wedgetype singularities, viz. peakons, are numerically identified bifurcating from smooth ground states. As anacademic case, we considered a perturbed version of thelocal formof the cdz equation, for which an analytical solution of peakons with exponential shape is derived.
14 F. FEDELE AND D. DUTYKH Figure. cdz envelope equation: Elastic collision of two traveling waves with a ground state (ε =.). Finally, by means of an accurate Fourier-type pseudo-spectral scheme, it is also shown that smooth solitary waves appear to collide elastically, suggesting the integrability of the compact Zakharov equation, but not that of the associated Hamiltonian version of the Dysthe equation. Acknowledgements D. Dutykh acknowledges the support from French Agence Nationale de la Recherche, project MathOcéan (Grant ANR-8-BLAN-3-). F. Fedele acknowledges the travel support received by the Geophysical Fluid Dynamics (GFD) Program to attend part of the summer school at the Woods Hole Oceanographic Institution in August and. F. Fedele thanks Prof. Gregory Eyink for useful discussions on weak solutions of partial differential equations. The authors are also grateful to Profs. Mike Banner, Didier Clamond, Taras Lakoba, Paul Milewski, and Jianke Yang for useful discussions on the subject of nonlinear waves.
15 SOLUTIONS TO A COMPACT DEEP-WATER EQUATION B Figure 3. cdz envelope equation: Initial () shape and () after collision of the three solitons of Figure. References [] M. J. Ablowitz, D. J. Kaup, A. C. Newell, and H. Segur. The inverse scattering transform-fourier analysis for nonlinear problems. Stud. Appl. Math., 53:9 35, 97. [] M. J. Ablowitz and H. Segur. Solitons and the Inverse Scattering Transform. Society for Industrial & Applied Mathematics, 98. [3] J. P. Boyd. Chebyshev and Fourier Spectral Methods. nd edition,. 6 [] R. Camassa and D. Holm. An integrable shallow water equation with peaked solitons. Phys. Rev. Lett., 7():66 66, 993., 6 [5] D. Clamond and J. Grue. A fast method for fully nonlinear water-wave computations. J. Fluid. Mech., 7: ,. [6] P. G. Drazin and R. S. Johnson. Solitons: An introduction. Cambridge University Press, Cambridge, 989. [7] A. I. Dyachenko and V. Zakharov. Is free-surfacehydrodynamics an integrable system? Physics Letters A, 9(): 8, 99. [8] A. I. Dyachenko and V. Zakharov. Toward an integrable model of deep water. Physics Letters A, (-):8 8, 996. [9] A. I. Dyachenko and V. E. Zakharov. Compact Equation for Gravity Waves on Deep Water. JETP Lett., 93():7 75,.,, 5, 7 [] A. I. Dyachenko, V. E. Zakharov, and D. I. Kachulin. Collision of two breathers at surface of deep water. ariv:.88, page 5,. 3 [] K. B. Dysthe. Note on a modification to the nonlinear Schrödinger equation for application to deep water. Proc. R. Soc. Lond. A, 369:5, 979., 5
16 6 F. FEDELE AND D. DUTYKH.6.5. B Figure. cdz-dysthe envelope equation: Initial () shape and () after collision of two traveling waves and a ground state (ε =.). [] F. Fedele and D. Dutykh. Hamiltonian form and solitary waves of the spatial Dysthe equations. JETP Lett., 9():8 8, October., 5, 6 [3] M. Frigo and S. G. Johnson. The Design and Implementation of FFTW3. Proceedings of the IEEE, 93():6 3, 5. 6 [] D. Fructus, D. Clamond, O. Kristiansen, and J. Grue. An efficient model for threedimensional surface wave simulations. Part I: Free space problems. J. Comput. Phys., 5: , 5. [5] O. Gramstad and K. Trulsen. Hamiltonian form of the modified nonlinear Schrödinger equation for gravity waves on arbitrary depth. J. Fluid Mech, 67: 6,. [6] S. J. Hogan. The fourth-order evolution equation for deep-water gravity-capillary waves. Proc. R. Soc. A, (83):359 37, 985. [7] V. P. Krasitskii. On reduced equations in the Hamiltonian theory of weakly nonlinear surface waves. J. Fluid Mech, 7:, 99. [8] T. I. Lakoba and J. Yang. A generalized Petviashvili iteration method for scalar and vector Hamiltonian equations with arbitrary form of nonlinearity. J. Comp. Phys., 6:668 69, 7. 3, 5 [9] M. S. Longuet-Higgins and J. H. Fox. Asymptotic theory for the almost-highest solitary wave. J. Fluid Mech, 37: 9, [] M. S. Longuet-Higgins and M. J. H. Fox. Theory of the almost-highest wave: the inner solution. J. Fluid Mech., 8:7 7, April [] M. S. Longuet-Higgins and M. J. H. Fox. Theory of the almost-highest wave. Part. Matching and analytic extension. Journal of Fluid Mechanics, 85: , April [] E. N. Lorenz. Energy and numerical weather prediction. Tellus, :36 373, 96. [3] P. Milewski and E. Tabak. A pseudospectral procedure for the solution of nonlinear wave equations with examples from free-surface flows. SIAM J. Sci. Comput., (3):, 999.
17 SOLUTIONS TO A COMPACT DEEP-WATER EQUATION 7 [] P. J. Morrison. Hamiltonian description of the ideal fluid. Rev. Mod. Phys., 7():67 5, 998., 5 [5] D. Pelinovsky and Y. A. Stepanyants. Convergence of Petviashvili s iteration method for numerical approximation of stationary solutions of nonlinear wave equations. SIAM J. Num. Anal., : 7,. 5 [6] V. I. Petviashvili. Equation of an extraordinary soliton. Sov. J. Plasma Phys., (3):69 7, , 5 [7] R. L. Seliger and G. B. Whitham. Variational principle in continuous mechanics. Proc. R. Soc. Lond. A, 35: 5, 968. [8] G. Söderlind. Digital filters in adaptive time-stepping. ACM Trans. Math. Software, 9: 6, 3. [9] G. Söderlind and L. Wang. Adaptive time-stepping and computational stability. Journal of Computational and Applied Mathematics, 85():5 3, 6. [3] M. Stiassnie. Note on the modified nonlinear Schrödinger equation for deep water waves. Wave Motion, 6():3 33, July 98. [3] M. Stiassnie and L. Shemer. On modifications of the Zakharov equation for surface gravity waves. J. Fluid Mech, 3:7 67, 98. [3] L. N. Trefethen. Spectral methods in MatLab. Society for Industrial and Applied Mathematics, Philadelphia, PA, USA,. 6, [33] K. Trulsen and K. B. Dysthe. Frequency downshift in three-dimensional wave trains in a deep basin. J. Fluid Mech, 35: , 997. [3] N. G. Vakhitov and A. A. Kolokolov. Stationary solutions of the wave equation in a medium with nonlinearity saturation. Radiophysics and Quantum Electronics, 6(7): , July [35] J. H. Verner. Explicit Runge-Kutta methods with estimates of the local truncation error. SIAM J. Num. Anal., 5():77 79, 978. [36] G. B. Whitham. Linear and nonlinear waves. John Wiley & Sons Inc., New York, 999. [37] J. Yang. Nonlinear Waves in Integrable and Nonintegrable Systems. SIAM,. 5, 6 [38] V. E. Zakharov. Stability of periodic waves of finite amplitude on the surface of a deep fluid. J. Appl. Mech. Tech. Phys., 9:9 9, 968., [39] V. E. Zakharov. Statistical theory of gravity and capillary waves on the surface of a finite-depth fluid. Eur. J. Mech. B/Fluids, 8(3):37 3, 999. [] V. E. Zakharov, P. Guyenne, A. N. Pushkarev, and F. Dias. Wave turbulence in one-dimensional models. Physica D, 53:573 69,. 6 [] V. E. Zakharov and A. B. Shabat. Exact Theory of Two-dimensional Self-focusing and Onedimensional Self-modulation of Waves in Nonlinear Media. Soviet Physics-JETP, 3:6 69, 97. School of Civil and Environmental Engineering & School of Electrical and Computer Engineering, Georgia Institute of Technology, Atlanta, USA address: fedele@gatech.edu URL: LAMA, UMR 57 CNRS, Université de Savoie, Campus Scientifique, Le Bourgetdu-Lac Cedex, France address: Denys.Dutykh@univ-savoie.fr URL:
arxiv: v1 [physics.flu-dyn] 14 Jun 2014
Observation of the Inverse Energy Cascade in the modified Korteweg de Vries Equation D. Dutykh and E. Tobisch LAMA, UMR 5127 CNRS, Université de Savoie, Campus Scientifique, 73376 Le Bourget-du-Lac Cedex,
More informationModel Equation, Stability and Dynamics for Wavepacket Solitary Waves
p. 1/1 Model Equation, Stability and Dynamics for Wavepacket Solitary Waves Paul Milewski Mathematics, UW-Madison Collaborator: Ben Akers, PhD student p. 2/1 Summary Localized solitary waves exist in the
More informationExponential asymptotics theory for stripe solitons in two-dimensional periodic potentials
NLS Workshop: Crete 2013 Exponential asymptotics theory for stripe solitons in two-dimensional periodic potentials Jianke Yang University of Vermont Collaborators: Sean Nixon (University of Vermont) Triantaphyllos
More informationPhysics Letters A 374 (2010) Contents lists available at ScienceDirect. Physics Letters A.
Physics Letters A 374 00) 4 45 Contents lists available at ScienceDirect Physics Letters A www.elsevier.com/locate/pla Variational derivation of improved KP-type of equations Lie She Liam a,, E. van Groesen
More informationAlgebraic geometry for shallow capillary-gravity waves
Algebraic geometry for shallow capillary-gravity waves DENYS DUTYKH 1 Chargé de Recherche CNRS 1 Université de Savoie Laboratoire de Mathématiques (LAMA) 73376 Le Bourget-du-Lac France Seminar of Computational
More informationOn deviations from Gaussian statistics for surface gravity waves
On deviations from Gaussian statistics for surface gravity waves M. Onorato, A. R. Osborne, and M. Serio Dip. Fisica Generale, Università di Torino, Torino - 10125 - Italy Abstract. Here we discuss some
More informationarxiv:nlin/ v1 [nlin.si] 25 Sep 2006
Remarks on the conserved densities of the Camassa-Holm equation Amitava Choudhuri 1, B. Talukdar 1a and S. Ghosh 1 Department of Physics, Visva-Bharati University, Santiniketan 73135, India Patha Bhavana,
More informationGeneration of undular bores and solitary wave trains in fully nonlinear shallow water theory
Generation of undular bores and solitary wave trains in fully nonlinear shallow water theory Gennady El 1, Roger Grimshaw 1 and Noel Smyth 2 1 Loughborough University, UK, 2 University of Edinburgh, UK
More informationFOURTH ORDER NONLINEAR EVOLUTION EQUATIONS FOR GRAVITY-CAPILLARY WAVES IN THE PRESENCE OF A THIN THERMOCLINE IN DEEP WATER
ANZIAM J. 43(2002), 513 524 FOURTH ORDER NONLINEAR EVOLUTION EQUATIONS FOR GRAVITY-CAPILLARY WAVES IN THE PRESENCE OF A THIN THERMOCLINE IN DEEP WATER SUMA DEBSARMA 1 andk.p.das 1 (Received 23 March 1999)
More informationOn the Whitham Equation
On the Whitham Equation Henrik Kalisch Department of Mathematics University of Bergen, Norway Joint work with: Handan Borluk, Denys Dutykh, Mats Ehrnström, Daulet Moldabayev, David Nicholls Research partially
More informationLecture 4: Birkhoff normal forms
Lecture 4: Birkhoff normal forms Walter Craig Department of Mathematics & Statistics Waves in Flows - Lecture 4 Prague Summer School 2018 Czech Academy of Science August 31 2018 Outline Two ODEs Water
More informationDerivation of Generalized Camassa-Holm Equations from Boussinesq-type Equations
Derivation of Generalized Camassa-Holm Equations from Boussinesq-type Equations H. A. Erbay Department of Natural and Mathematical Sciences, Faculty of Engineering, Ozyegin University, Cekmekoy 34794,
More informationEvolution of a nonlinear wave field along a tank: experiments and numerical simulations based on the spatial Zakharov equation
J. Fluid Mech. (), vol. 7, pp. 7 9. Printed in the United Kingdom c Cambridge University Press 7 Evolution of a nonlinear wave field along a tank: experiments and numerical simulations based on the spatial
More informationFAST COMMUNICATION THREE-DIMENSIONAL LOCALIZED SOLITARY GRAVITY-CAPILLARY WAVES
COMM. MATH. SCI. Vol. 3, No., pp. 89 99 c 5 International Press FAST COMMUNICATION THREE-DIMENSIONAL LOCALIZED SOLITARY GRAVITY-CAPILLARY WAVES PAUL A. MILEWSKI Abstract. In a weakly nonlinear model equation
More informationLecture17: Generalized Solitary Waves
Lecture17: Generalized Solitary Waves Lecturer: Roger Grimshaw. Write-up: Andrew Stewart and Yiping Ma June 24, 2009 We have seen that solitary waves, either with a pulse -like profile or as the envelope
More informationAn integrable shallow water equation with peaked solitons
An integrable shallow water equation with peaked solitons arxiv:patt-sol/9305002v1 13 May 1993 Roberto Camassa and Darryl D. Holm Theoretical Division and Center for Nonlinear Studies Los Alamos National
More informationRelation between Periodic Soliton Resonance and Instability
Proceedings of Institute of Mathematics of NAS of Ukraine 004 Vol. 50 Part 1 486 49 Relation between Periodic Soliton Resonance and Instability Masayoshi TAJIRI Graduate School of Engineering Osaka Prefecture
More informationFreedom in the Expansion and Obstacles to Integrability in Multiple-Soliton Solutions of the Perturbed KdV Equation
Freedom in the Expansion and Obstacles to Integrability in Multiple-Soliton Solutions of the Perturbed KdV Equation Alex Veksler 1 and Yair Zarmi 1, Ben-Gurion University of the Negev, Israel 1 Department
More informationNotes on the Inverse Scattering Transform and Solitons. November 28, 2005 (check for updates/corrections!)
Notes on the Inverse Scattering Transform and Solitons Math 418 November 28, 2005 (check for updates/corrections!) Among the nonlinear wave equations are very special ones called integrable equations.
More informationModeling and predicting rogue waves in deep water
Modeling and predicting rogue waves in deep water C M Schober University of Central Florida, Orlando, Florida - USA Abstract We investigate rogue waves in the framework of the nonlinear Schrödinger (NLS)
More informationDiagonalization of the Coupled-Mode System.
Diagonalization of the Coupled-Mode System. Marina Chugunova joint work with Dmitry Pelinovsky Department of Mathematics, McMaster University, Canada Collaborators: Mason A. Porter, California Institute
More informationOn weakly singular and fully nonlinear travelling shallow capillary gravity waves in the critical regime
arxiv:1611.115v3 [physics.class-ph] 8 Mar 17 Dimitrios Mitsotakis Victoria University of Wellington, New Zealand Denys Dutykh CNRS, Université Savoie Mont Blanc, France Aydar Assylbekuly Khoja Akhmet Yassawi
More informationWeak Turbulence of Gravity Waves
JETP Letters, Vol. 77, No. 0, 003, pp. 546 550. From Pis ma v Zhurnal Éksperimental noœ i Teoreticheskoœ Fiziki, Vol. 77, No. 0, 003, pp. 649 653. Original English Text Copyright 003 by Dyachenko, Korotkevich,
More informationModified Serre Green Naghdi equations with improved or without dispersion
Modified Serre Green Naghdi equations with improved or without dispersion DIDIER CLAMOND Université Côte d Azur Laboratoire J. A. Dieudonné Parc Valrose, 06108 Nice cedex 2, France didier.clamond@gmail.com
More informationA Note On Solitary Wave Solutions of the Compound Burgers-Korteweg-de Vries Equation
A Note On Solitary Wave Solutions of the Compound Burgers-Korteweg-de Vries Equation arxiv:math/6768v1 [math.ap] 6 Jul 6 Claire David, Rasika Fernando, and Zhaosheng Feng Université Pierre et Marie Curie-Paris
More informationDavydov Soliton Collisions
Davydov Soliton Collisions Benkui Tan Department of Geophysics Peking University Beijing 100871 People s Republic of China Telephone: 86-10-62755041 email:dqgchw@ibmstone.pku.edu.cn John P. Boyd Dept.
More informationStable One-Dimensional Dissipative Solitons in Complex Cubic-Quintic Ginzburg Landau Equation
Vol. 112 (2007) ACTA PHYSICA POLONICA A No. 5 Proceedings of the International School and Conference on Optics and Optical Materials, ISCOM07, Belgrade, Serbia, September 3 7, 2007 Stable One-Dimensional
More informationStatistical properties of mechanically generated surface gravity waves: a laboratory experiment in a 3D wave basin
Statistical properties of mechanically generated surface gravity waves: a laboratory experiment in a 3D wave basin M. Onorato 1, L. Cavaleri 2, O.Gramstad 3, P.A.E.M. Janssen 4, J. Monbaliu 5, A. R. Osborne
More informationUniversity of Bath. DOI: /j.wavemoti Publication date: Document Version Peer reviewed version. Link to publication
Citation for published version: Kim, B, Dias, F & Milewski, PA 01, 'On weakly nonlinear gravity capillary solitary waves' Wave Motion, vol. 49, no., pp. 1-37. https://doi.org/10.1016/j.wavemoti.011.10.00
More informationLecture 10: Whitham Modulation Theory
Lecture 10: Whitham Modulation Theory Lecturer: Roger Grimshaw. Write-up: Andong He June 19, 2009 1 Introduction The Whitham modulation theory provides an asymptotic method for studying slowly varying
More informationRELAXED VARIATIONAL PRINCIPLE FOR WATER WAVE MODELING
RELAXED VARIATIONAL PRINCIPLE FOR WATER WAVE MODELING DENYS DUTYKH 1 Senior Research Fellow UCD & Chargé de Recherche CNRS 1 University College Dublin School of Mathematical Sciences Workshop on Ocean
More informationLecture 7: Oceanographic Applications.
Lecture 7: Oceanographic Applications. Lecturer: Harvey Segur. Write-up: Daisuke Takagi June 18, 2009 1 Introduction Nonlinear waves can be studied by a number of models, which include the Korteweg de
More informationNumerical study of the generalised Klein-Gordon equations
Denys Dutykh CNRS LAMA, University Savoie Mont Blanc, France Marx Chhay LOCIE, University Savoie Mont Blanc, France Didier Clamond University of Nice Sophia Antipolis, France arxiv:138.2521v2 [physics.class-ph]
More informationExact and approximate nonlinear waves generated by the periodic superposition of solitons
Journal of Applied Mathematics and Physics (ZAMP) 0044-2275/89/060940-05 $ 1.50 + 0.20 Vol. 40, November 1989 9 1989 Birkhguser Verlag, Basel Exact and approximate nonlinear waves generated by the periodic
More informationNote on Breather Type Solutions of the NLS as Models for Freak-Waves
Physica Scripta. Vol. T8, 48^5, 1999 Note on Breather Type Solutions of the NLS as Models for Freak-Waves Kristian B. Dysthe 1 and Karsten Trulsen y 1 Department of Mathematics, University of Bergen, Johs.Brunsgt.1,
More informationA Low-Dimensional Model for the Maximal Amplification Factor of Bichromatic Wave Groups
PROC. ITB Eng. Science Vol. 35 B, No., 3, 39-53 39 A Low-Dimensional Model for the Maximal Amplification Factor of Bichromatic Wave Groups W. N. Tan,* & Andonowati Fakulti Sains, Universiti Teknologi Malaysia
More informationExperimental and numerical study of spatial and temporal evolution of nonlinear wave groups
Nonlin. Processes Geophys., 15, 91 942, 28 www.nonlin-processes-geophys.net/15/91/28/ Author(s) 28. This work is distributed under the Creative Commons Attribution. License. Nonlinear Processes in Geophysics
More informationLong-time solutions of the Ostrovsky equation
Long-time solutions of the Ostrovsky equation Roger Grimshaw Centre for Nonlinear Mathematics and Applications, Department of Mathematical Sciences, Loughborough University, U.K. Karl Helfrich Woods Hole
More informationNonlinear Waves, Solitons and Chaos
Nonlinear Waves, Solitons and Chaos Eryk Infeld Institute for Nuclear Studies, Warsaw George Rowlands Department of Physics, University of Warwick 2nd edition CAMBRIDGE UNIVERSITY PRESS Contents Foreword
More informationSimulations and experiments of short intense envelope
Simulations and experiments of short intense envelope solitons of surface water waves A. Slunyaev 1,), G.F. Clauss 3), M. Klein 3), M. Onorato 4) 1) Institute of Applied Physics, Nizhny Novgorod, Russia,
More informationNUMERICAL SOLITARY WAVE INTERACTION: THE ORDER OF THE INELASTIC EFFECT
ANZIAM J. 44(2002), 95 102 NUMERICAL SOLITARY WAVE INTERACTION: THE ORDER OF THE INELASTIC EFFECT T. R. MARCHANT 1 (Received 4 April, 2000) Abstract Solitary wave interaction is examined using an extended
More informationWhat Is a Soliton? by Peter S. Lomdahl. Solitons in Biology
What Is a Soliton? by Peter S. Lomdahl A bout thirty years ago a remarkable discovery was made here in Los Alamos. Enrico Fermi, John Pasta, and Stan Ulam were calculating the flow of energy in a onedimensional
More informationThe effect of a background shear current on large amplitude internal solitary waves
The effect of a background shear current on large amplitude internal solitary waves Wooyoung Choi Dept. of Mathematical Sciences New Jersey Institute of Technology CAMS Report 0506-4, Fall 005/Spring 006
More informationLecture 12: Transcritical flow over an obstacle
Lecture 12: Transcritical flow over an obstacle Lecturer: Roger Grimshaw. Write-up: Erinna Chen June 22, 2009 1 Introduction The flow of a fluid over an obstacle is a classical and fundamental problem
More informationA new integrable system: The interacting soliton of the BO
Phys. Lett., A 204, p.336-342, 1995 A new integrable system: The interacting soliton of the BO Benno Fuchssteiner and Thorsten Schulze Automath Institute University of Paderborn Paderborn & Germany Abstract
More informationarxiv:patt-sol/ v1 25 Sep 1995
Reductive Perturbation Method, Multiple Time Solutions and the KdV Hierarchy R. A. Kraenkel 1, M. A. Manna 2, J. C. Montero 1, J. G. Pereira 1 1 Instituto de Física Teórica Universidade Estadual Paulista
More informationSpatial evolution of an initially narrow-banded wave train
DOI 10.1007/s40722-017-0094-6 RESEARCH ARTICLE Spatial evolution of an initially narrow-banded wave train Lev Shemer 1 Anna Chernyshova 1 Received: 28 February 2017 / Accepted: 26 July 2017 Springer International
More informationThe Nonlinear Schrodinger Equation
Catherine Sulem Pierre-Louis Sulem The Nonlinear Schrodinger Equation Self-Focusing and Wave Collapse Springer Preface v I Basic Framework 1 1 The Physical Context 3 1.1 Weakly Nonlinear Dispersive Waves
More informationNonlinear Fourier Analysis
Nonlinear Fourier Analysis The Direct & Inverse Scattering Transforms for the Korteweg de Vries Equation Ivan Christov Code 78, Naval Research Laboratory, Stennis Space Center, MS 99, USA Supported by
More informationA small dispersion limit to the Camassa Holm equation: A numerical study
Available online at www.sciencedirect.com Mathematics and Computers in Simulation 80 (2009) 120 130 A small dispersion limit to the Camassa Holm equation: A numerical study Jennifer Gorsky a,, David P.
More informationMechanisms of Interaction between Ultrasound and Sound in Liquids with Bubbles: Singular Focusing
Acoustical Physics, Vol. 47, No., 1, pp. 14 144. Translated from Akusticheskiœ Zhurnal, Vol. 47, No., 1, pp. 178 18. Original Russian Text Copyright 1 by Akhatov, Khismatullin. REVIEWS Mechanisms of Interaction
More informationStrongly nonlinear long gravity waves in uniform shear flows
Strongly nonlinear long gravity waves in uniform shear flows Wooyoung Choi Department of Naval Architecture and Marine Engineering, University of Michigan, Ann Arbor, Michigan 48109, USA Received 14 January
More informationThree-dimensional gravity-capillary solitary waves in water of finite depth and related problems. Abstract
Three-dimensional gravity-capillary solitary waves in water of finite depth and related problems E.I. Părău, J.-M. Vanden-Broeck, and M.J. Cooker School of Mathematics, University of East Anglia, Norwich,
More informationEvolution of kurtosis for wind waves
GEOPHYSICAL RESEARCH LETTERS, VOL. 36, L13603, doi:10.1029/2009gl038613, 2009 Evolution of kurtosis for wind waves S. Y. Annenkov 1 and V. I. Shrira 1 Received 14 April 2009; revised 21 May 2009; accepted
More informationProgress In Electromagnetics Research Letters, Vol. 10, 69 75, 2009 TOPOLOGICAL SOLITONS IN 1+2 DIMENSIONS WITH TIME-DEPENDENT COEFFICIENTS
Progress In Electromagnetics Research Letters, Vol. 10, 69 75, 2009 TOPOLOGICAL SOLITONS IN 1+2 DIMENSIONS WITH TIME-DEPENDENT COEFFICIENTS B. Sturdevant Center for Research and Education in Optical Sciences
More informationBasic Concepts and the Discovery of Solitons p. 1 A look at linear and nonlinear signatures p. 1 Discovery of the solitary wave p. 3 Discovery of the
Basic Concepts and the Discovery of Solitons p. 1 A look at linear and nonlinear signatures p. 1 Discovery of the solitary wave p. 3 Discovery of the soliton p. 7 The soliton concept in physics p. 11 Linear
More informationNo-hair and uniqueness results for analogue black holes
No-hair and uniqueness results for analogue black holes LPT Orsay, France April 25, 2016 [FM, Renaud Parentani, and Robin Zegers, PRD93 065039] Outline Introduction 1 Introduction 2 3 Introduction Hawking
More informationChapter 3. Head-on collision of ion acoustic solitary waves in electron-positron-ion plasma with superthermal electrons and positrons.
Chapter 3 Head-on collision of ion acoustic solitary waves in electron-positron-ion plasma with superthermal electrons and positrons. 73 3.1 Introduction The study of linear and nonlinear wave propagation
More informationIntegrable dynamics of soliton gases
Integrable dynamics of soliton gases Gennady EL II Porto Meeting on Nonlinear Waves 2-22 June 213 Outline INTRODUCTION KINETIC EQUATION HYDRODYNAMIC REDUCTIONS CONCLUSIONS Motivation & Background Main
More informationThe effect of third-order nonlinearity on statistical properties of random directional waves in finite depth
Nonlin. Processes Geophys., 16, 11 19, 29 www.nonlin-processes-geophys.net/16/11/29/ Author(s) 29. This work is distributed under the Creative Commons Attribution. License. Nonlinear Processes in Geophysics
More informationOccurrence of Freak Waves from Envelope Equations in Random Ocean Wave Simulations
Occurrence of Freak Waves from Envelope Equations in Random Ocean Wave Simulations Miguel Onorato, Alfred R. Osborne, Marina Serio, and Tomaso Damiani Universitá di Torino, Via P. Giuria, - 025, Torino,
More informationBoussineq-Type Equations and Switching Solitons
Proceedings of Institute of Mathematics of NAS of Ukraine, Vol. 3, Part 1, 3 351 Boussineq-Type Equations and Switching Solitons Allen PARKER and John Michael DYE Department of Engineering Mathematics,
More informationarxiv: v1 [nlin.cd] 21 Mar 2012
Approximate rogue wave solutions of the forced and damped Nonlinear Schrödinger equation for water waves arxiv:1203.4735v1 [nlin.cd] 21 Mar 2012 Miguel Onorato and Davide Proment Dipartimento di Fisica,
More informationBäcklund transformation and special solutions for Drinfeld Sokolov Satsuma Hirota system of coupled equations
arxiv:nlin/0102001v1 [nlin.si] 1 Feb 2001 Bäcklund transformation and special solutions for Drinfeld Sokolov Satsuma Hirota system of coupled equations Ayşe Karasu (Kalkanli) and S Yu Sakovich Department
More informationSurface Tension Effect on a Two Dimensional. Channel Flow against an Inclined Wall
Applied Mathematical Sciences, Vol. 1, 007, no. 7, 313-36 Surface Tension Effect on a Two Dimensional Channel Flow against an Inclined Wall A. Merzougui *, H. Mekias ** and F. Guechi ** * Département de
More informationModulation and water waves
Summer School on Water Waves................Newton Institute August 2014 Part 1 Thomas J. Bridges, University of Surrey Modulation of uniform flows and KdV h 0 u 0 00000000000000000000000000000 11111111111111111111111111111
More informationCity, University of London Institutional Repository
City Research Online City, University of London Institutional Repository Citation: Wang, J., Ma, Q. & Yan, S. (2017). On quantitative errors of two simplified unsteady models for simulating unidirectional
More informationLecture 1: Water waves and Hamiltonian partial differential equations
Lecture : Water waves and Hamiltonian partial differential equations Walter Craig Department of Mathematics & Statistics Waves in Flows Prague Summer School 208 Czech Academy of Science August 30 208 Outline...
More informationCircular dispersive shock waves in colloidal media
University of Wollongong Research Online Faculty of Engineering and Information Sciences - Papers: Part B Faculty of Engineering and Information Sciences 6 Circular dispersive shock waves in colloidal
More informationMoving Weakly Relativistic Electromagnetic Solitons in Laser-Plasmas
Moving Weakly Relativistic Electromagnetic Solitons in Laser-Plasmas Lj. Hadžievski, A. Mančić and M.M. Škorić Department of Physics, Faculty of Sciences and Mathematics, University of Niš, P.O. Box 4,
More informationFinding eigenvalues for matrices acting on subspaces
Finding eigenvalues for matrices acting on subspaces Jakeniah Christiansen Department of Mathematics and Statistics Calvin College Grand Rapids, MI 49546 Faculty advisor: Prof Todd Kapitula Department
More informationPeriodic oscillations in the Gross-Pitaevskii equation with a parabolic potential
Periodic oscillations in the Gross-Pitaevskii equation with a parabolic potential Dmitry Pelinovsky 1 and Panos Kevrekidis 2 1 Department of Mathematics, McMaster University, Hamilton, Ontario, Canada
More informationExact Solutions to the Focusing Discrete Nonlinear Schrödinger Equation
Exact Solutions to the Focusing Discrete Nonlinear Schrödinger Equation Francesco Demontis (based on a joint work with C. van der Mee) Università degli Studi di Cagliari Dipartimento di Matematica e Informatica
More informationMesoscopic Wave Turbulence
JETP Letters, Vol. 8, No. 8, 005, pp. 487 49. From Pis ma v Zhurnal Éksperimental noœ i Teoreticheskoœ Fiziki, Vol. 8, No. 8, 005, pp. 544 548. Original English Text Copyright 005 by Zakharov, Korotkevich,
More informationA Dimension-Breaking Phenomenon for Steady Water Waves with Weak Surface Tension
A Dimension-Breaking Phenomenon for Steady Water Waves with Weak Surface Tension Erik Wahlén, Lund University, Sweden Joint with Mark Groves, Saarland University and Shu-Ming Sun, Virginia Tech Nonlinear
More informationThe General Form of Linearized Exact Solution for the KdV Equation by the Simplest Equation Method
Applied and Computational Mathematics 015; 4(5): 335-341 Published online August 16 015 (http://www.sciencepublishinggroup.com/j/acm) doi: 10.11648/j.acm.0150405.11 ISSN: 38-5605 (Print); ISSN: 38-5613
More informationExperiments on extreme wave generation using the Soliton on Finite Background
Experiments on extreme wave generation using the Soliton on Finite Background René H.M. Huijsmans 1, Gert Klopman 2,3, Natanael Karjanto 3, and Andonawati 4 1 Maritime Research Institute Netherlands, Wageningen,
More information1 Introduction. A. M. Abourabia 1, K. M. Hassan 2, E. S. Selima 3
ISSN 1749-3889 (print), 1749-3897 (online) International Journal of Nonlinear Science Vol.9(2010) No.4,pp.430-443 The Derivation and Study of the Nonlinear Schrödinger Equation for Long Waves in Shallow
More informationProgramming of the Generalized Nonlinear Paraxial Equation for the Formation of Solitons with Mathematica
American Journal of Applied Sciences (): -6, 4 ISSN 546-99 Science Publications, 4 Programming of the Generalized Nonlinear Paraxial Equation for the Formation of Solitons with Mathematica Frederick Osman
More informationRogue waves in large-scale fully-non-linear High-Order-Spectral simulations
Rogue waves in large-scale fully-non-linear High-Order-Spectral simulations Guillaume Ducrozet, Félicien Bonnefoy & Pierre Ferrant Laboratoire de Mécanique des Fluides - UMR CNRS 6598 École Centrale de
More informationCONSTRUCTION OF THE HALF-LINE POTENTIAL FROM THE JOST FUNCTION
CONSTRUCTION OF THE HALF-LINE POTENTIAL FROM THE JOST FUNCTION Tuncay Aktosun Department of Mathematics and Statistics Mississippi State University Mississippi State, MS 39762 Abstract: For the one-dimensional
More informationBifurcations of solitons and their stability
1 Bifurcations of solitons and their stability E. A. Kuznetsov 1,2,3 & F. Dias 4,5 1 P.N. Lebedev Physical Institute, 119991 Moscow, Russia 2 L.D. Landau Institute for Theoretical Physics, 119334 Moscow,
More informationHamiltonian dynamics of breathers with third-order dispersion
1150 J. Opt. Soc. Am. B/ Vol. 18, No. 8/ August 2001 S. Mookherjea and A. Yariv Hamiltonian dynamics of breathers with third-order dispersion Shayan Mookherjea* and Amnon Yariv Department of Applied Physics
More informationA Numerical Solution of the Lax s 7 th -order KdV Equation by Pseudospectral Method and Darvishi s Preconditioning
Int. J. Contemp. Math. Sciences, Vol. 2, 2007, no. 22, 1097-1106 A Numerical Solution of the Lax s 7 th -order KdV Equation by Pseudospectral Method and Darvishi s Preconditioning M. T. Darvishi a,, S.
More informationExact Travelling Wave Solutions of the Coupled Klein-Gordon Equation by the Infinite Series Method
Available at http://pvamu.edu/aam Appl. Appl. Math. ISSN: 93-9466 Vol. 6, Issue (June 0) pp. 3 3 (Previously, Vol. 6, Issue, pp. 964 97) Applications and Applied Mathematics: An International Journal (AAM)
More informationThe 3-wave PDEs for resonantly interac7ng triads
The 3-wave PDEs for resonantly interac7ng triads Ruth Mar7n & Harvey Segur Waves and singulari7es in incompressible fluids ICERM, April 28, 2017 What are the 3-wave equa.ons? What is a resonant triad?
More informationSeparatrix Map Analysis for Fractal Scatterings in Weak Interactions of Solitary Waves
Separatrix Map Analysis for Fractal Scatterings in Weak Interactions of Solitary Waves By Yi Zhu, Richard Haberman, and Jianke Yang Previous studies have shown that fractal scatterings in weak interactions
More informationKeywords: Exp-function method; solitary wave solutions; modified Camassa-Holm
International Journal of Modern Mathematical Sciences, 2012, 4(3): 146-155 International Journal of Modern Mathematical Sciences Journal homepage:www.modernscientificpress.com/journals/ijmms.aspx ISSN:
More informationWeak Turbulence in Ocean Waves
Weak Turbulence in Ocean Waves Yuri V. Lvov Rensselaer Polytechnic Institute Troy, New York 12180-3590 Phone: (518) 276-6893 fax: (518) 276-4824 email: lvov@rpi.edu Award Number N000140210528 http://www.rpi.edu/~lvovy
More informationReview of Fundamental Equations Supplementary notes on Section 1.2 and 1.3
Review of Fundamental Equations Supplementary notes on Section. and.3 Introduction of the velocity potential: irrotational motion: ω = u = identity in the vector analysis: ϕ u = ϕ Basic conservation principles:
More informationUndular Bores and the Morning Glory
Undular Bores and the Morning Glory Noel F. Smyth School of Mathematics and Mawell Institute for Mathematical Sciences, The King s Buildings, University of Edinburgh, Edinburgh, Scotland, U.K., EH9 3JZ.
More informationarxiv:math/ v1 [math.ap] 1 Jan 1992
APPEARED IN BULLETIN OF THE AMERICAN MATHEMATICAL SOCIETY Volume 26, Number 1, Jan 1992, Pages 119-124 A STEEPEST DESCENT METHOD FOR OSCILLATORY RIEMANN-HILBERT PROBLEMS arxiv:math/9201261v1 [math.ap]
More informationBreather propagation in shallow water. 1 Introduction. 2 Mathematical model
Breather propagation in shallow water O. Kimmoun 1, H.C. Hsu 2, N. Homann 3,4, A. Chabchoub 5, M.S. Li 2 & Y.Y. Chen 2 1 Aix-Marseille University, CNRS, Centrale Marseille, IRPHE, Marseille, France 2 Tainan
More informationLecture 15: Waves on deep water, II
Lecture 15: Waves on deep water, II Lecturer: Harvey Segur. Write-up: Andong He June 23, 2009 1 Introduction. In the previous lecture (Lecture 14) we sketched the derivation of the nonlinear Schrödinger
More information2. The generalized Benjamin- Bona-Mahony (BBM) equation with variable coefficients [30]
ISSN 1749-3889 (print), 1749-3897 (online) International Journal of Nonlinear Science Vol.12(2011) No.1,pp.95-99 The Modified Sine-Cosine Method and Its Applications to the Generalized K(n,n) and BBM Equations
More informationA numerical study of the long wave short wave interaction equations
Mathematics and Computers in Simulation 74 007) 113 15 A numerical study of the long wave short wave interaction equations H. Borluk a, G.M. Muslu b, H.A. Erbay a, a Department of Mathematics, Isik University,
More informationComputational Solutions for the Korteweg devries Equation in Warm Plasma
COMPUTATIONAL METHODS IN SCIENCE AND TECHNOLOGY 16(1, 13-18 (1 Computational Solutions for the Korteweg devries Equation in Warm Plasma E.K. El-Shewy*, H.G. Abdelwahed, H.M. Abd-El-Hamid. Theoretical Physics
More informationTravelling Wave Solutions for the Gilson-Pickering Equation by Using the Simplified G /G-expansion Method
ISSN 1749-3889 (print, 1749-3897 (online International Journal of Nonlinear Science Vol8(009 No3,pp368-373 Travelling Wave Solutions for the ilson-pickering Equation by Using the Simplified /-expansion
More informationLecture 13: Triad (or 3-wave) resonances.
Lecture 13: Triad (or 3-wave) resonances. Lecturer: Harvey Segur. Write-up: Nicolas Grisouard June 22, 2009 1 Introduction The previous lectures have been dedicated to the study of specific nonlinear problems,
More informationB 2 P 2, which implies that g B should be
Enhanced Summary of G.P. Agrawal Nonlinear Fiber Optics (3rd ed) Chapter 9 on SBS Stimulated Brillouin scattering is a nonlinear three-wave interaction between a forward-going laser pump beam P, a forward-going
More information